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I've been struggling with translating random slope and intercept and random variables and understanding them as latent variables in the pursuit of path models.

For example here is a random slope and random intercept model of say students nested in schools. Using hierarchal notation: $$ Y_{ij} = \beta_{0j} + \beta_{1j} X_{ij} + e_{ij} $$

$$ \beta_{0j} = \gamma_{00} + U_{0j} $$

$$ \beta_{1j} = \gamma_{10} + U_{1j} $$

Translating this into path diagram I have latent variables the $U_{0j}$ random intercept and $U_{1j}$ to indicate the random slope. I am not sure how to connect them to other parts of the diagram in the student level.

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From my reading there are two reasons to convert a hlm into a path diagram. For a visual representation of the model that is seen (Curran & Bauer, 2007). My interests were in the estimation of SEM. (Paras & Neale, 2005) offers this guidance.

Following guidance the random intercept and slope $U_{0j}$ and $U_{1j}$ are circles to represent latent variables. $X$ and $Y$ are manifest variables and in boxes. The one triangle represents the intercept. I denoted with greek symbols what each path is estimating. How $U_{0j}$ connects with $Y$ is with a fixed value of 1. $U_{1j}$ connects with $Y$ with the value of $X$. Paras & Neale refer to this as a definition variable. The factor loading is the observation's value of $X$.

enter image description here

For how to estimate this model in OpenMx see: How to fit random slope hierarchical model as SEM with OpenMx in R?

References

Mehta, P. D., & Neale, M. C. (2005). People are variables too: multilevel structural equations modeling. Psychological methods, 10(3), 259.

Curran, P. J., & Bauer, D. J. (2007). Building path diagrams for multilevel models. Psychological Methods, 12(3), 283.

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